Behavior of Shared Suction Anchors in Clay Overlying Silty Sand Soils Considering the Souring Effect

Abstract

This paper investigates, through finite element analysis, the bearing capacity behavior of the shared suction anchors in clay-covered silt soil layers, considering the effects of soil scour. Its aim is to address the anchors’ failure mechanisms and corresponding bearing capacity. The numerical model was validated against previously reported data, with good agreement obtained. The main findings are as follows: (1) the tensional force T exerts an influence on the horizontal bearing capacity; (2) it is proven that scour significantly affects the failure mechanism of the suction anchor in clay overlying silty sand and, consequently, the corresponding bearing capacity; and (3) a bearing capacity design process for the shared suction anchor subjected to combined VHMT loading in clay-covered silt soil layers, considering soil scour, is proposed to provide guidance for practical engineering.

1. Introduction

1.1. Background

In marine energy, offshore wind power is developing rapidly and has become an important driver of energy transformation and of achieving carbon peaking and carbon neutrality goals [1]. Suction anchors are a common foundation in marine engineering [2]. A typical suction anchor is a large-diameter cylinder with an opening at the bottom and a bulge on its skirt. It is connected to floating bodies via mooring lines and is commonly used on coastal floating platforms, such as offshore floating wind turbines and deep-water projects. As shown in Figure 1a, the shared anchor is a new technology which reduces the cost of the foundation. This technology primarily uses a single suction anchor or piled anchor to connect multiple anchor chains, thereby connecting multiple floating wind turbines and reducing the number of suction anchors required for production and installation, thus reducing costs.

Shared suction anchor has good performance in reducing the cost of the foundation for offshore wind turbines. However, there remains uncertainty in its design, especially in predicting the bearing capacity under combined loads, considering scour effects.

1.2. Previous Work

Research on shared suction anchors began relatively late as an emerging technology in offshore engineering. Existing studies primarily focus on conceptual design and anchor chain force analysis, while limited attention has been paid to their VHMT combined bearing capacity in specific layered soils—especially under scouring conditions—which limits their engineering application. For the fundamental theory of suction anchor bearing capacity, research on marine VHMT bearing capacity has advanced through numerical and analytical methods, extending the failure envelope from 3D to 4D and providing theoretical support for offshore foundation design. He & Newson (2023) derived a 4D undrained failure envelope for two-layer clay shallow foundations with crust-correction factors, cautioning against capacity underestimation without crust consideration [3]. Liu et al. (2023) found that torsion weakens the VHM capacity of suction caissons (L/D = 1.0–2.0) in normally consolidated clay (more significantly for smaller L/D) and proposed a torsion-included calculation method [4]. Fu et al. (2024) demonstrated that fins enhance the bearing capacities of suction caissons in clay, analyzed key influencing factors, and established closed-form expressions for three torsion scenarios [5].

Specific to shared suction anchors, early explorations have focused on mooring system optimization: Fontana et al. (2016) [6] proposed the shared mooring concept for floating offshore wind turbines, showing that anchor point numbers can be reduced by at least threefold; Balakrishnan et al. (2020) [7] simulated shared anchor loading using a 5 MW wind turbine floating system; Saviano et al. (2017) [8] conducted finite element analysis of shared suction anchors’ non-displacement high-pressure performance in soft soil.

To address the above research gaps, this study investigates the VHMT combined bearing capacity characteristics of shared suction anchors in layered clay-silt soils under scouring conditions, aiming to provide targeted design guidance for their offshore engineering applications.

1.3. Objective of the Present Study

This study investigates the bearing characteristics of shared suction anchors under VHMT loads and soil scour conditions. The failure mechanisms and corresponding envelope curves for shared suction anchors were proposed, considering the soil scouring issue, which can provide guidance for the construction and design of shared suction anchors in offshore wind turbines.

2. Numerical Model

2.1. Geometric Modeling of Shared Suction Anchor

Based on the research by Zhou et al. (2023) and its application in practical engineering, the shared suction anchor involved in this study is cylindrical, with a diameter of D and a length of L [9]. The length-to-diameter ratio of the shared suction anchor ranges from 1 to 3. The material of the shared suction anchor is modeled as a rigid body using a C3D8 mesh, and the external load application point is set to the bottom-center reference point of the shared suction anchor.

2.2. Geometric Modeling of Soil Mass

The soil in this study consisted of a clay-over-silt layer. The clay was modeled as an elastoplastic material, following the Mohr–Coulomb failure criterion, and classified as a normally consolidated soil. A C3D8 mesh was used, with a Poisson’s ratio of υ = 0.49, and the friction angle and dilation angle set to 0. An undrained condition model with no volume change was employed. The clay layer thickness was T_c, and the clay layer thickness ratio T_c/D ranged from 0.5 to 2. The undrained shear strength S_u as a function of depth was expressed as follows:

S_u = S_um + kz

(1)

where S_um represents the undrained shear strength of the clay at the mud surface; k is the gradient of the clay shear strength with depth; and z is the depth of the soil below the seabed. The elastic modulus E is related to S_u by E = 500S_u. In this study, the normalized parameter S_um/(kD) represents the clay strength, and the change in this parameter is used as a parameter analysis of the clay strength, where D is the diameter of the shared suction anchor. According to the study by Zhou et al. (2023) [9], the values of S_um/kD are set to 0, 0.67, and 1.0, which are the effective unit weights of the clay. Detailed parameters of the clay are shown in

Table 1. Clay modeling parameters.

Parameter	Clay1	Clay2	Clay3
Sum (kPa)	0.1	5	10
k (kPa/m)	1.25	1.5	2
${\mathit{\gamma }}_{\mathbf{c}}\mathit{\prime }$ (kN/m3)	6	6	6
Sum/(kD)	0	0.67	1.0

[9].

The underlying silt was modeled using the Mohr–Coulomb failure criterion, and the soil was modeled using a C3D8 mesh. Draw on existing research data about silt (Zhou et al., 2023), the cohesion C_u of the silt ranged from 3 to 15 kPa, and its normalized cohesion was expressed as denoted by $C_{\mathrm{u}}/({{\gamma }_{\mathrm{s}}}^{\prime }D)$, where ${{\gamma }_{\mathrm{s}}}^{\prime }$ is the effective unit weight of the silt, D is the diameter of the shared suction anchor, and takes values of 0.07, 0.22, and 0.34 [9]. The friction angle φ_s of the silt was set to 33°, and the dilation angle ψ was set to 8°. Since the ultimate bearing capacity of sandy soil mainly depends on the soil strength, the elastic modulus of the silt needs to be set sufficiently large. The elastic modulus of the silt was greater than 2000 C_u. Detailed parameters of the silt are shown in

Table 2. Modeling parameters for silt and sand.

Parameter	Silt1	Silt2	Silt3
Cu (kPa)	3	10	15
${\mathit{\phi }}_{\mathit{s}}$(°)	33	33	33
$\mathit{\psi }$ (°)	8	8	8
${{\mathit{\gamma }}_{\mathbf{s}}}^{\mathit{\prime }}$(kN/m3)	8.9	8.9	8.9
${\mathit{C}}_{\mathbf{u}}\mathbf{/}\mathbf{\left(}{{\mathit{\gamma }}_{\mathbf{s}}}^{\prime }\mathit{D}\mathbf{\right)}$	0.07	0.22	0.34

.

The local scour effect manifests as soil loss around the suction anchor in soil modeling. The scour hole is usually idealized as an inverted cone, defined by the scour depth S_d, scour hole angle φ, and bottom width S_wb, as illustrated in Figure 1. Based on existing research (Guo et al., 2022) and offshore engineering practices, the bottom width of the scour hole S_wb in this study is recommended to be set to 0, and the scour hole angle φ is suggested to be fixed at 30° [10]. Therefore, the scour depth S_d is adopted as the key parameter for the scour hole in this research, with a value range of 0 to 1D, resulting in a ratio of scour depth to the diameter of the shared suction anchor S_d/D ranging from 0 to 1. It should be noted that this method simplifies the local scour hole, which may lead to deviations from practical engineering conditions. In actual engineering applications, adjustments to the scour angle and slope stability should be made based on site-specific construction conditions.

The soil is modeled as a cylindrical shape. To eliminate boundary effects, the soil diameter is set to 15D = 75 m, and the soil depth is set to L + 4.5D [11]. The mesh is refined from the edge of the soil to the center and from the top and bottom of the soil to the loading point of the suction anchor, as shown in Figure 2

2.3. Combination of Soil Mass and Shared Suction Anchors

A shared suction anchor is installed at the center of the soil mass, with its base embedded in the silty sand layer. The anchor and soil assembly are allowed to separate. The interface between the soil and the anchor follows Coulomb’s law of friction, with the friction coefficient denoted by μ. The friction coefficient between the anchor and clay is μ = 1. To account for the impact of the anchor installation on the soil, a thin layer of soil, 0.2 m thick, is placed outside the anchor in the clay layer. Its undrained shear strength is defined as 0.65S_u [12]. Therefore, the equivalent fractional coefficient is 1 × 0.65 = 0.65. The interaction between the anchor and silty sand is μ = 0.4. The loading point at the center of the anchor’s bottom is coupled to the anchor. The bottom of the soil mass is completely fixed, and horizontal displacement is constrained around the perimeter of the soil mass.

2.4. Model Validation

For suction anchors installed in clay, this study employs numerical simulation results for suction anchors in clay and compares them with published data reported by Zhou et al. (2023) [9]. As illustrated in Figure 3a, the VH envelope curve of the proposed model is compared with existing findings, revealing good agreement between the two. Consequently, the accuracy of the clay model established in this study is validated.

For suction anchors installed in silty sand, this study conducts a comparative analysis by adopting the numerical simulation and centrifuge test results for suction anchors in silty sand presented by Kim et al. (2015) [13]. As depicted in Figure 3b, the load–displacement curves of the proposed model under horizontal loading are compared with the existing data, demonstrating a good agreement between them. Thus, the accuracy of the silty sand model developed in this study is verified.

To verify the accuracy of the model considering soil scour effects in this study, the simulation results modeled above were compared with the numerical simulation results of suction anchors considering soil scour effects by Guo et al. (2022), as shown in Figure 3 [10]. The uniaxial loading results for V, H, and M modeled in this study were compared one by one with the results of Guo et al. (2022) (L/D = 2) [10]. The results are in good agreement, indicating that the accuracy of the model considering soil scour has been verified.

3. Soil Failure Mechanism

In practical marine geotechnical engineering, soil scour is often a key factor to consider [14]. The main varying parameters are the scour depth ratio S_d/D, the length-to-diameter ratio of the suction anchor L/D, the clay strength S_um/(kD), the clay layer thickness ratio T_c/D, and the silt cohesion $C_{\mathrm{u}}/({\gamma }_{\mathrm{s}}\prime D)$. The shared suction anchor is subjected to uniaxial, biaxial, and quadriaxial loading. The ultimate bearing capacities V₀, H₀, M₀, and T₀ of the shared suction anchor under uniaxial loading (V–H, H–M, H–T), considering scour effects, are obtained, as well as the failure envelope curves for biaxial loading (V–H, H–M, H–T) and quadriaxial loading (VHMT). The calculation results are presented in

Table 3. Calculation table for shared suction anchors considering soil scour.

Examples	Sum/(kD)	${\mathit{C}}_{\mathbf{u}}/({{\mathit{\gamma }}_{\mathbf{s}}}^{\prime }\mathit{D})$	Tc/D	L/D	Sd/D	V/V0	T/T0	Loading Methods
1	-	-	-	-	-	-	-	Model Validation
2	0–1	0.07–0.34	0.5–1.5	1–3	0–1	-	-	Uniaxial Loading
3a	0–1	0.07–0.34	0.5–1.5	1–3	0–1	-	-	V-H Combined Loading
3b	0–1	0.07–0.34	0.5–1.5	1–3	0–1	-	-	H-M Combined Loading
3c	0–1	0.07–0.34	0.5–1.5	1–3	0–1	-	-	H-T Combined Loading
4	0–1	0.07–0.34	0.5–1.5	1–3	0–1	0.5–0.9	0.5–0.9	VHMT Combined Loading

.

To illustrate the failure mechanisms of a shared suction anchor in a clay-covered silt layer considering soil scour, taking soil parameters S_um/(kD) = 0, $C_{\mathrm{u}}/({{\gamma }_{\mathrm{s}}}^{\prime }D)$ = 0.34, T_c/D = 1, L/D = 3, S_d/D = 0.5 as an example, the displacement vector diagram of its single-axis loading of V, H, M, and T is shown in Figure 4. The bottom of the suction anchor is located in the silty sand. In Figure 4a, under horizontal loading, without soil scour, the rotation of the suction anchor causes the soil on the left surface to be squeezed upward and undergo shear failure. Under the action of soil scour, the rotation of the suction anchor causes the soil on the left to be squeezed downward and undergo shear failure, and the soil on the right to be squeezed upward and undergo shear failure. In Figure 4b, under bending-moment loading, considering soil scour, the suction anchor rotates and squeezes the soil on the right side, causing shear failure downwards. Therefore, soil scour will change the direction of shear failure of the surface soil under H and M loading [15]. In Figure 4c,d, under V and T loading, the soil failure mechanism is shear failure caused by friction at the contact surface between the soil and the suction anchor [16]. As illustrated in Figure 4a,b, scouring induces the removal of substantial clay around the anchor, which in turn weakens the lateral bearing capacity of the anchor shaft and results in a slight downward shift in the rotation center.

4. Loading Methods

4.1. Uniaxial Loading

The uniaxial loading method for the shared suction anchor is displacement-controlled loading. Displacements in the corresponding directions V, H, M, and T are applied at the loading point of the shared suction anchor, and the displacements are gradually increased until the single-bearing load tends to stabilize, thus obtaining the ultimate uniaxial bearing capacity of the shared suction anchor.

4.1.1. Uniaxial V Loading

The uniaxial V-load was controlled by displacement. An upward displacement was applied at the loading point to obtain the uniaxial ultimate bearing capacity V₀, which was then normalized to obtain v₀. Figure 5 shows the uniaxial bearing capacity v₀. In Figure 5a, L/D = 1, S_d/D takes values of 0, 0.5, 1, and T_c/D = 0.5. According to Equation (2),

$$v_0={\displaystyle \frac{V_0}{\left(\frac{T_{\mathrm{c}}{\gamma }_{\mathrm{c}}\prime +\left(L-T_{\mathrm{c}}\right){{\gamma }_{\mathrm{s}}}^{\prime }}{L}\right)D^3}}$$

(2)

as S_d/D increases, the uniaxial bearing capacity v₀ gradually decreases. This is because the increased size of the scour hole leads to a reduction in the soil mass around the suction anchor, decreasing the contact area between the suction anchor and the soil, thus reducing the frictional force and consequently decreasing v₀. Simultaneously, the figure shows that when S_d/D = 0 (i.e., no soil scour), the change in the soil parameter clay strength S_um/(kD) has the most significant impact on v₀. When S_d/D increases to 1, the change in S_um/(kD) has no significant effect on v₀. Therefore, the reduction in soil mass weakens the influence of soil parameter changes on the uniaxial bearing capacity of the suction anchor. The increase in silty sand cohesion $C_{\mathrm{u}}/({{\gamma }_{\mathrm{s}}}^{\prime }D)$ has a relatively weak effect on v₀.

In Figure 5b,c, S_d/D takes values of 0.5 and 1, and the clay layer thickness ratio T_c/D takes values of 1 and 1.5. In Figure 5b, L/D = 1.5, and in Figure 5c, L/D = 3. It can be seen from the figures that increasing T_c/D leads to a decrease in v₀, while increasing S_d/D weakens the effect of T_c/D on v₀, because the larger the scour hole, the less soil is affected. Comparing Figure 5b,c, increasing L/D significantly increases v₀, and the increase in L/D weakens the effect of S_d/D on v₀, indicating that the greater the suction anchor burial depth, the less the bearing capacity is affected by changes in the scour hole. The increase in silt cohesion $C_{\mathrm{u}}/({\gamma }_{\mathrm{s}}\prime D)$ has a weak effect on v₀. Referring to existing research [10], the scour factor ${\eta }_{\mathrm{V}}$ is used to represent the ratio of vertical bearing capacity under scour conditions to vertical bearing capacity under non-scour conditions, as shown in

Table 4. Vertical bearing capacity scour factor.

Tc/D	L/D	Sd/D	${\mathit{\eta }}_{\mathbf{V}}$
0.5	1	0.5	0.75
1	1.5	0.5	0.88
1.5	1.5	0.5	0.95
1	3	0.5	0.92
1.5	3	0.5	0.89
0.5	1	1	0.22
1	1.5	1	0.70
1.5	1.5	1	0.79
1	3	1	0.84
1.5	3	1	0.79

. The calculation method for the scour factor η is shown in Equation (3).

$$\mathsf{\eta }=F_{ult,s}/F_{ult,0}$$

(3)

where F_ult,s: ultimate uniaxial bearing capacity under scoured conditions (including vertical ultimate bearing capacity F_V,ult,s, horizontal ultimate bearing capacity F_H,ult,s, and moment ultimate bearing capacity F_M,ult,s).

F_ult,0: corresponding ultimate uniaxial bearing capacity under unscoured conditions.

4.1.2. Uniaxial H Loading

Uniaxial H-loading was performed using displacement control. A horizontal displacement was applied at the loading point to obtain the uniaxial horizontal ultimate bearing capacity H₀, which was then dimensionless converted to h₀. The loading results are shown in Figure 6. In Figure 6a, increasing S_d/D decreases h₀, with h₀ being minimum when S_d/D = 1. Furthermore, the influence of changes in clay strength and silt cohesion on h₀ is at its weakest; h₀ increases with increasing silt cohesion.

As shown in Figure 6b,c, when T_c/D increases, h₀ decreases, and the change in clay strength S_um/(kD) has a stronger impact on h₀. In Figure 6b, when T_c/D increases to 1.5, T_c/D = L/D, at which point h₀ is at its minimum, and the influence of silt cohesion on horizontal bearing capacity is weakest. An increase in S_d/D reduces horizontal bearing capacity because the reduction in soil mass decreases the resistance from the soil during the horizontal displacement of the suction anchor, thus reducing h₀. Furthermore, an increase in S_d/D weakens the influence of soil strength parameters and T_c/D on horizontal bearing capacity. An increase in the suction anchor’s length-to-diameter ratio L/D can weaken the influence of S_d/D on horizontal bearing capacity. The scour factor ${\eta }_{\mathrm{H}}$ is used to represent the ratio of horizontal bearing capacity under scour conditions to that under non-scour conditions, as shown in

Table 5. Horizontal bearing capacity scour factor.

Tc/D	L/D	Sd/D	${\mathit{\eta }}_{\mathbf{H}}$
0.5	1	0.5	0.66
1	1.5	0.5	0.75
1.5	1.5	0.5	0.85
1	3	0.5	0.87
1.5	3	0.5	0.87
0.5	1	1	0.19
1	1.5	1	0.55
1.5	1.5	1	0.64
1	3	1	0.64
1.5	3	1	0.69

.

4.1.3. Uniaxial M Loading

The uniaxial M-load is displacement-controlled, with a rotation angle applied at the loading point to obtain the uniaxial bending moment ultimate bearing capacity, which is then dimensionless processed to obtain m₀. Figure 7 shows the loading results. In Figure 7, L/D = 1. Increasing S_d/D leads to a decrease in m₀ and also weakens the influence of changes in clay strength and silt cohesion on m₀. The bending moment bearing capacity of the suction anchor is the weakest, and the influence of clay strength and silt cohesion on m₀ is also the weakest.

As shown in Figure 7b,c, the increase in S_d/D significantly reduces m0 because the reduction in soil mass decreases the resistance encountered when the suction anchor overturns. This principle is similar to the effect of S_d/D on h₀. The increase in the suction anchor’s length-to-diameter ratio L/D weakens the effect of S_d/D on the bending moment bearing capacity. m₀ increases with the increase in clay strength and silt cohesion, and gradually decreases with the increase in T_c/D. When S_d/D increases, the effects of clay strength, silt cohesion, and T_c/D on the bending moment bearing capacity weaken. The scour factor ${\eta }_{\mathrm{M}}$ is used to represent the ratio of the bending moment bearing capacity under scour conditions to that under non-scour conditions, as shown in

Table 6. Bending moment bearing capacity scour factor.

Tc/D	L/D	Sd/D	${\mathit{\eta }}_{\mathbf{M}}$
0.5	1	0.5	0.59
1	1.5	0.5	0.68
1.5	1.5	0.5	0.66
1	3	0.5	0.67
1.5	3	0.5	0.7
0.5	1	1	0.14
1	1.5	1	0.36
1.5	1.5	1	0.38
1	3	1	0.67
1.5	3	1	0.66

.

4.1.4. Uniaxial T Loading

Uniaxial T loading employed displacement control, applying a torsional rotation angle at the loading point to obtain the uniaxial torque ultimate bearing capacity, which was then dimensionless processed to obtain t₀. The loading results are shown in Figure 8. Since the resistance experienced by the suction anchor during torsion is friction, similar to uniaxial V-loading, t₀ gradually decreases with increasing T_c/D, increases with increasing clay strength S_um/(kD), and decreases with increasing S_d/D. Increasing L/D increases t₀ and weakens the effect of scouring on the torque bearing capacity. The figure shows that t₀ increases with increasing silt cohesion. The scouring factor ${\eta }_{\mathrm{T}}$ is used to represent the ratio of the torque bearing capacity under scouring to that under no-scouring conditions, as shown in

Table 7. Torque bearing capacity erosion factor.

Tc/D	L/D	Sd/D	${\mathit{\eta }}_{\mathbf{T}}$
0.5	1	0.5	0.85
1	1.5	0.5	0.67
1.5	1.5	0.5	0.87
1	3	0.5	0.86
1.5	3	0.5	0.86
0.5	1	1	0.22
1	1.5	1	0.49
1.5	1.5	1	0.65
1	3	1	0.84
1.5	3	1	0.81

. As indicated in Table 4, Table 5, Table 6 and Table 7, soil scour induces a reduction in the single-anchor bearing capacity of the shared suction anchor, with the decreases in h0 and m0 being particularly significant—this is consistent with the research findings of Guo et al. (2022) [10]. As S_d/D increases, the single-anchor bearing capacity decreases; while the magnitude of the numerical influence of changes in clay strength S_um/(kD) and the ratio of silty sand cohesion to clay layer thickness T_c/D on the single-anchor bearing capacity diminishes, the underlying influence trends remain unchanged. An increase in the suction anchor’s length-to-diameter ratio L/D can mitigate the adverse effects caused by the increase in S_d/D and simultaneously significantly enhance the single-anchor bearing capacity. Therefore, this study concludes that L/D is the primary influencing factor governing the single-anchor bearing capacity of the suction anchor. Notably, when T_c/D = L/D, the suction anchor is installed at the interface between clay and sand, resulting in a marked reduction in the single-anchor bearing capacity.

4.2. Biaxial Combined Loading

The biaxial combined loading method is employed to study the bearing capacity of the shared suction anchor, undertaken through displacement control for the load application. Specifically, displacements in V–H, H–M, and H–T directions are simultaneously imposed. The displacement ratio is a constant while displacements are gradually increased until the bearing capacity stabilizes. This stable value is then recorded to derive the biaxial stress–failure envelope curve of the shared suction anchor.

4.2.1. V-H Combined Loading

Figure 9 shows the VH envelope curve of the bearing capacity of the shared suction caisson. As observed, v first increases and then decreases with increasing h, reaching its maximum value near 0.5h₀. This is because its ultimate uplift capacity increases with the increase in horizontal load, resulting in v > v₀. As the horizontal load further increases and h approaches approximately 0.5h₀, v rises to its maximum value. With a continued increase in the horizontal load until h = h₀, v gradually decreases to 0. In Figure 9a,b, the increase in S_um/(kD) and $C_{\mathrm{u}}/({\gamma }_{\mathrm{s}}\prime D)$ leads to a higher VH envelope value, indicating an enhanced bearing capacity of the suction anchor. In Figure 9c, the increase in T_c/D leads to a decrease in the VH envelope value because of the increased proportion of clay in the soil; hence, it reduces the equivalent soil strength in the pre-embedded length of the caisson. Consequently, the suction anchor bearing capacity drops accordingly. In Figure 9d, the increase in L/D of the caisson results in an effect on the failure mechanism of the caisson, which increases the V–H envelope value. Figure 9e,f show that the VH envelope value decreases with the scour depth increase because the effective caisson’s length is decreased due to soil scour, thereby reducing the suction anchor’s bearing capacity. In Figure 9e, for the case of L/D = 3, the shape of the VH envelope does not change with the increase in scour depth. In Figure 9f, for the case of L/D = 1, the shape of the VH envelope changes significantly with the increase in scour depth, and the envelope value decreases significantly. This indicates that when L/D is small, the scour depth has a greater impact on the bearing capacity and envelope shape of the suction anchor. However, as L/D increases, the impact of scour depth on the bearing capacity and envelope shape of the suction anchor weakens. Therefore, increasing the length-to-diameter ratio of the suction anchor can reduce the impact of soil scour.

4.2.2. H–M Combined Loading

Figure 10 shows the HM envelope curve. As can be seen, the curve is eccentric to the right. After increasing to its maximum value, the bending moment m decreases and eventually reduces to 0, which is consistent with the research findings of Hung and Kim (2014) [17]. In Figure 10, for the case of S_d/D = 1, as $C_{\mathrm{u}}/({\gamma }_{\mathrm{s}}\prime D)$ increases, the value of the envelope curve increases. In Figure 10b, the value of the envelope curve decreases as T_c/D increases, in which the suction anchor’s load-bearing capacity decreases accordingly. In Figure 10c,d, the value of the envelope curve increases as S_um/(kD) increases, but the shape does not change significantly. Compared with that in Figure 10c,d, it can be seen that increasing the length-to-diameter ratio L/D of the suction anchor significantly increases the value of the envelope curve. And the larger the length-to-diameter ratio, the more pronounced the eccentricity of the curve. Figure 10e,f show the effect of the scour depth ratio S_d/D on the HM envelope. It shows that the value of the envelope decreases as S_d/D increases. For the case of L/D = 1, the scour depth has a greater impact on the value, while for that of L/D = 3, the impact of the scour depth on the value weakens. When S_d/D increases, the eccentricity of the envelope increases, which is consistent with the conclusions of Guo et al. (2022) [10].

4.2.3. H–T Combined Loading

As shown in Figure 11, the HT envelope curve first increases and then decreases to 0 as h increases. When h is near 0.75h₀, t increases to its maximum value. This is because its ultimate uplift capacity increases with the horizontal load increase, resulting in t > t₀. As the horizontal load further increases and h approaches approximately 0.75h₀, t rises to its maximum value. With a continued increase in the horizontal load until h = h₀, t gradually decreases to 0. In Figure 11a,b, the value of the envelope increases with the increase in clay strength and silty sand’s cohesion, while its shape remains largely unchanged. In Figure 11c, the increase in the clay layer thickness ratio T_c/D causes the envelope value to decrease, while the shape of the envelope remains unchanged. In Figure 11d, the value of the envelope increases significantly with the increase in L/D, and the ratio of the maximum value of t to t₀ increases, while the shape of the envelope remains unchanged. In Figure 11e,f, the values of the envelope decrease as S_d/D increases. When S_d/D increases, the ratio of the maximum value of t to t₀ decreases. In Figure 11f, L/D = 1, the increase in scour depth has a significant impact on the bearing capacity of the suction anchor. In Figure 11e, for the case of L/D = 3, the increase in scour depth has a weaker impact on the bearing capacity of the suction anchor.

4.3. VHMT Combined Loading

(1)

Vertical Loading (V). In this analysis step, a concentrated upward vertical force V is applied at the center loading point at the bottom of the shared suction anchor in the global coordinate system. The range of V is 0.3V₀–0.9V₀, where V₀ is the ultimate uniaxial bearing capacity in the vertical direction.

(2)

Torque Loading (T). In this analysis step, a torque T is applied at the loading point of the shared suction anchor in the global coordinate system. The range of T is 0.3T₀–0.9T₀, where T₀ is the ultimate torque of the individually applied torque.

(3)

Displacement Loading (H-M). In this analysis step, a horizontal displacement U1 and a rotational displacement UR2 are applied at the loading point of the shared suction anchor in the global coordinate system. The loading is adjusted by controlling the displacement ratio.

4.3.1. Influence of Changes in V/V₀ and T/T₀

Figure 12 shows the effect of changes in V/V₀ and T/T₀ on the bearing capacity of the shared suction anchor VHMT, with identical V/V₀ and T/T₀ fixed as 0.5, 0.7, and 0.9. In Figure 12a,b, for the case of S_d/D = 0.5, V/V₀ increases from 0.5 to 0.9, and T/T₀ remains constant at 0.5. It can be seen from the figure that the envelope shrinks with the increase in V/V₀. The larger V/V₀ is, the more obvious the envelope shrinkage is. In fact, as V increases, the shrinkage trend of the envelope becomes more pronounced, leading to greater reductions in both the horizontal bearing capacity and the bending moment bearing capacity. This is because, as V increases and approaches V₀, only a small additional force is required to induce failure. Consequently, both the horizontal bearing capacity and the moment bearing capacity exhibit a decreasing trend. This indicates that the soil scouring effect does not change the trend of the influence of V/V₀ on the bearing capacity of the suction anchor VHMT.

In Figure 12c,d, for the case of S_d/D = 0.5, T/T₀ increases from 0.5 to 0.9, and V/V₀ remains unchanged at 0.5. It shows that the envelope value decreases with T/T₀ increases, the curve contracts, and the curve shape does not change significantly. As T increases, the shrinkage of the envelope becomes more pronounced, resulting in lower horizontal bearing capacity and bending moment bearing capacity. Moreover, the larger T/T₀ is, the greater the decrease in the envelope value. This indicates that the soil scouring effect has not changed the trend of the influence of T/T₀ on the bearing capacity of the suction anchor VHMT. In fact, under unscoured conditions, the increasing trends in Figure 12e,f are similar to those in Figure 12a,b, and the increasing trends in Figure 12g,h are analogous to those in Figure 12c,d.

4.3.2. Influence of Changes in Soil Parameters

Figure 13 illustrates the impact of soil parameter variations on the bearing capacity of the shared suction anchor VHMT. In Figure 13a,b, for the case of S_d/D = 0.5, and S_um/(kD) increases from 0 to 1. In Figure 13c,d, $C_{\mathrm{u}}/({\gamma }_{\mathrm{s}}\prime D)$ increases from 0.07 to 0.34. Both increases lead to an increase in the envelope value, enhancing the suction anchor’s bearing capacity, and causing the curve to expand outward. This indicates that the soil scouring effect does not alter the trend of soil strength variation on the VHMT’s bearing capacity.

In Figure 13e,f, for the case of S_d/D = 0.5, and T_c/D takes values of 1 and 1.5. The figures show that increasing T_c/D significantly reduces the envelope value. Consequently, it decreases the suction anchor’s bearing capacity, causing the curve to contract significantly. This is because as T_c/D increases, the equivalent soil strength along the caisson is reduced, hence affecting the normalized bearing capacity of the anchor.

4.3.3. Influence of Changes in the Aspect Ratio of Suction Anchors

Figure 14 illustrates the effect of varying the length-to-diameter ratio of the suction anchor on the VHMT bearing capacity of the shared suction anchor. In Figure 14, L/D values are 1.5 and 3. As can be seen in the figure, with L/D increases, the envelope value increases significantly, and the eccentricity of the curve increases. In Figure 14b, L/D values are 1 and 1.5. When L/D increases, the envelope value increases, the VHMT bearing capacity of the shared suction anchor is enhanced, and the eccentricity of the envelope curve increases.

4.3.4. Influence of Changes in Scour Depth

The effect of scour depth variation on the VHMT bearing capacity of the shared suction anchor is shown in Figure 15. In Figure 15a, for the case of L/D = 3 with S_d/D varied as 0, 0.5, and 1, it can be seen from the figure that the increase in scour depth reduces the value of the envelope curve, weakens the VHMT bearing capacity of the shared suction anchor, and increases the eccentricity of the curve. In Figure 15b, L/D = 1.5, the increase in scour depth leads to a greater reduction in the bearing capacity of the shared suction anchor, and the eccentricity of the envelope curve increases more significantly. In Figure 15c, L/D = 1, the increase in scour depth leads to a significant reduction in the VHMT bearing capacity of the shared suction anchor, and the eccentricity of the envelope curve increases significantly. Therefore, the smaller the length-to-diameter ratio, the more significant the effect of scour depth variation on the VHMT bearing capacity of the shared suction anchors. Conversely, increasing the length-to-diameter ratio can weaken the bearing capacity effect brought about by soil scour. This is attributed to the fact that soil scour strips off the shallow clay layer, impairing the lateral confinement and shear resistance of the upper part of the suction anchor and thereby inducing inward migration of the upper failure plane. In contrast, the deep, stiff silt layer remains intact, sustaining the outward extension tendency of the lower failure plane. Such asymmetric evolution of the failure plane deflects the resultant reaction force of the soil mass from the central axis of the suction anchor, ultimately leading to an increase in eccentricity.

5. Proposed Design Procedure

5.1. Bearing Capacity of a Single Bearing

Referring to existing research results [10], the scour factor η is used to quantitatively represent the influence of soil scour on the single bearing load capacity of a caisson. Then, Equations (2)–(10) are proposed to represent the single bearing load considering the soil scour effect, in which the coefficients for each formula are shown in

Table 8. Vertical load bearing capacity coefficient table.

Tc/D	L/D	a	b	c
0.5	1	0.79	0.07	−0.02
1	1.5	1.62	0.26	0.12
1	3	5.00	0.41	−0.19
1	3.5	6.18	0.43	−0.25
1.5	1.5	0.92	0.81	0.08
1.5	3	4.61	0.84	−0.17
1.5	3.5	5.75	0.99	−0.18
2	3	4.14	1.27	−0.06
2	3.5	5.34	0.97	−0.27
3	3.5	4.45	1.59	0.03

,

Table 9. Horizontal load bearing capacity coefficient table.

Tc/D	L/D	a	b	c
0.5	1	4.63	0.75	1.98
1	1.5	7.58	1.78	2.17
1	3	34.40	6.74	9.02
1	3.5	50.22	5.73	10.78
1.5	1.5	2.86	1.25	0.60
1.5	3	26.39	5.60	4.52
1.5	3.5	39.87	7.58	7.97
2	3	21.45	5.03	5.03
2	3.5	30.10	7.49	4.88
3	3.5	17.47	6.62	2.98

,

Table 10. Bending moment bearing capacity coefficient table.

Tc/D	L/D	a	b	c
0.5	1	1.68	0.50	0.69
1	1.5	2.98	1.45	0.92
1	3	39.47	6.43	22.59
1	3.5	61.04	9.61	27.18
1.5	1.5	1.23	0.90	0.23
1.5	3	23.25	4.27	8.00
1.5	3.5	48.69	7.89	22.05
2	3	11.59	5.11	2.49
2	3.5	26.26	6.25	9.78
3	3.5	10.10	7.40	1.00

and

Table 11. Torque bearing capacity coefficient table.

Tc/D	L/D	a	b	c
0.5	1	0.47	0.10	0.05
1	1.5	1.36	0.40	0.32
1	3	2.75	0.13	0.28
1	3.5	3.05	0.29	0.27
1.5	1.5	0.33	0.68	0.05
1.5	3	2.40	0.32	0.12
1.5	3.5	2.96	0.23	0.20
2	3	2.00	0.51	0.10
2	3.5	2.67	0.44	0.25
3	3.5	2.08	1.11	0.40

. The following expressions are derived by fitting the FE results from the parametric study:

$$v_0={\eta }_{\mathrm{V}}\left[a+b\times {\displaystyle \frac{S_{\mathrm{um}}}{kD}}+c\times \log \left({\displaystyle \frac{C_{\mathrm{u}}}{{{\gamma }_{\mathrm{s}}}^{\prime }D}}\right)\right]$$

(4)

$$h_0={\displaystyle \frac{H_0}{\left(\frac{T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+\left(L-T_{\mathrm{c}}\right){{\gamma }_{\mathrm{s}}}^{\prime }}{L}\right)D^3}}$$

(5)

$$h_0={\eta }_{\mathrm{H}}\left[a+b\times {\displaystyle \frac{S_{\mathrm{um}}}{kD}}+c\times \log \left({\displaystyle \frac{C_{\mathrm{u}}}{{{\gamma }_{\mathrm{s}}}^{\prime }D}}\right)\right]$$

(6)

$$m_0={\displaystyle \frac{M_0}{\left(\frac{T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+\left(L-T_{\mathrm{c}}\right){{\gamma }_{\mathrm{s}}}^{\prime }}{L}\right)D^4}}$$

(7)

$$m_0={\eta }_{\mathrm{M}}\left[a+b\times {\displaystyle \frac{S_{\mathrm{um}}}{kD}}+c\times \log \left({\displaystyle \frac{C_{\mathrm{u}}}{{{\gamma }_{\mathrm{s}}}^{\prime }D}}\right)\right]$$

(8)

$$t_0={\displaystyle \frac{T_0}{\left(\frac{T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+\left(L-T_{\mathrm{c}}\right){{\gamma }_{\mathrm{s}}}^{\prime }}{L}\right)D^4}}$$

(9)

$$t_0={\eta }_{\mathrm{T}}\left[a+b\times {\displaystyle \frac{S_{\mathrm{um}}}{kD}}+c\times \log \left({\displaystyle \frac{C_{\mathrm{u}}}{{{\gamma }_{\mathrm{s}}}^{\prime }D}}\right)\right]$$

(10)

where L/D = 1~3, T_c/D = 0.5~1.5, S_um/(kD) = 0~1, $C_{\mathrm{u}}/({{\gamma }_{\mathrm{s}}}^{\prime }D)$ = 0.07~0.34, S_d/D = 0~1, the scouring factor η for different scouring depth ratios is shown in

Table 12. The scouring factor corresponding to S_d/D = 0.5.

Tc/D	L/D	${\mathit{\eta }}_{\mathit{V}}$	${\mathit{\eta }}_{\mathit{H}}$	${\mathit{\eta }}_{\mathit{M}}$	${\mathit{\eta }}_{\mathit{T}}$
0.5	1	0.75	0.66	0.59	0.85
1	1.5	0.88	0.75	0.68	0.67
1.5	1.5	0.95	0.85	0.66	0.87
1	3	0.92	0.87	0.67	0.86
1.5	3	0.89	0.87	0.7	0.86

and

Table 13. The scouring factor corresponding to S_d/D = 1.

Tc/D	L/D	${\mathit{\eta }}_{\mathit{V}}$	${\mathit{\eta }}_{\mathit{H}}$	${\mathit{\eta }}_{\mathit{M}}$	${\mathit{\eta }}_{\mathit{T}}$
0.5	1	0.22	0.19	0.14	0.22
1	1.5	0.70	0.55	0.36	0.49
1.5	1.5	0.79	0.64	0.38	0.65
1	3	0.84	0.64	0.67	0.84
1.5	3	0.79	0.69	0.66	0.81

. The fitting effect of the single bearing capacity is shown in Figure 16.

5.2. VHMT Bearing Capacity Curve

First of all, the H₀ and M₀ obtained from uniaxial loading are denoted by H_u and M_u, respectively. As shown in Figure 12e–h, the values of H₀ and M₀ from the VHMT failure envelopes under different V/V₀ and T/T₀ loading combinations are extracted under unscoured conditions. H₀/H_u and M₀/M_u are calculated, respectively, to obtain the horizontal bearing capacity reduction coefficient and bending moment bearing capacity reduction coefficient of the shared suction anchor under different V/V₀ and T/T₀ loading, L/D, T_c/D, S_um/(kD) and $C_{\mathrm{u}}/({{\gamma }_{\mathrm{s}}}^{\prime }D)$ are orthogonalized, the average undrained shear strength of clay is denoted as S_u,avg, the effective unit weight of clay is denoted as ${{\gamma }_{\mathrm{c}}}^{\prime }$, the effective unit weight of silt is denoted as ${{\gamma }_{\mathrm{s}}}^{\prime }$, the friction angle of silt is ${\phi }_{\mathrm{s}}$, and cohesion of silt, denoted as C_u, to obtain the parameters T_c/L and ${\displaystyle \frac{S_{\mathrm{u},\mathrm{avg}}}{C_{\mathrm{u}}+\left(T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+L{{\gamma }_{\mathrm{s}}}^{\prime }\right)\tan {\phi }_{\mathrm{s}}}}$, and the least squares method is used to fit to obtain Equations (11) and (12):

$${\displaystyle \frac{H_0}{H_{\mathrm{u}}}}=1-e{\left({\displaystyle \frac{V}{V_0}}\right)}^{1.65}-0.24{\left({\displaystyle \frac{T}{T_0}}\right)}^{0.01}$$

(11)

$${\displaystyle \frac{M_0}{M_{\mathrm{u}}}}=1-z{\left({\displaystyle \frac{V}{V_0}}\right)}^{0.06}-0.13{\left({\displaystyle \frac{T}{T_0}}\right)}^{2.31}$$

(12)

where e and z are the correction coefficients of T_c/L, S_d/D, and ${\displaystyle \frac{S_{\mathrm{u},\mathrm{avg}}}{C_{\mathrm{u}}+\left(T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+L{{\gamma }_{\mathrm{s}}}^{\prime }\right)\tan {\phi }_{\mathrm{s}}}}$ the calculation results are shown in Equations (13) and (14).

$$e=\left(0.12{\displaystyle \frac{S_{\mathrm{u},\mathrm{avg}}}{C_{\mathrm{u}}+\left(T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+L{{\gamma }_{\mathrm{s}}}^{\prime }\right)\tan {\phi }_{\mathrm{s}}}}+0.65\right)\left(0.56{\displaystyle \frac{T_{\mathrm{c}}}{L}}+0.22\right)\left(0.06{\displaystyle \frac{S_d}{D}}+0.33\right)$$

(13)

$$z=\left(0.3{\left({\displaystyle \frac{S_{\mathrm{u},\mathrm{avg}}}{C_{\mathrm{u}}+\left(T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+L{{\gamma }_{\mathrm{s}}}^{\prime }\right)\tan {\phi }_{\mathrm{s}}}}\right)}^{0.4}+0.44\right)\left(0.69{\displaystyle \frac{T_{\mathrm{c}}}{L}}+0.66\right)\left(0.12{\displaystyle \frac{S_d}{D}}+0.34\right)$$

(14)

The second step is to normalize the VHMT envelope curve. The influence of each factor is shown in Figure 17.

The third step is to fit the normalized VHMT envelope curve, and Equation (15) is proposed using finite element numerical simulation as an alternative to traditional experiments:

$$f_{\mathrm{VHMT}}={\left({\displaystyle \frac{H}{H_0}}\right)}^2+{\left({\displaystyle \frac{M}{M_0}}\right)}^2-\alpha \beta \gamma \xi \sigma \left({\displaystyle \frac{H}{H_0}}\right)\left({\displaystyle \frac{M}{M_0}}\right)-1=0$$

(15)

Among them, it can be obtained from Equations (2)–(15). Using the orthogonalization process in the first step, α, β, γ, ξ, and σ are the correction coefficients of V/V₀, T/T₀, ${\displaystyle \frac{S_{\mathrm{u},\mathrm{avg}}}{C_{\mathrm{u}}+\left(T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+L{{\gamma }_{\mathrm{s}}}^{\prime }\right)\tan {\phi }_{\mathrm{s}}}}$, T_c/L, and S_d/D, respectively. α, β, γ, ξ, σ are obtained by fitting the control variables with the least squares method and are expressed by Equations (15)–(20):

$$\alpha =0.72{\left({\displaystyle \frac{V}{V_0}}\right)}^{0.22}+0.68$$

(16)

$$\beta =0.2{\left({\displaystyle \frac{T}{T_0}}\right)}^{2.99}+0.74$$

(17)

$$\gamma =0.12{\left({\displaystyle \frac{S_{\mathrm{u},\mathrm{avg}}}{C_{\mathrm{u}}+\left(T_{\mathrm{c}}{{\gamma }_{\mathrm{c}}}^{\prime }+L{{\gamma }_{\mathrm{s}}}^{\prime }\right)\tan {\phi }_{\mathrm{s}}}}\right)}^{2.89}+0.65$$

(18)

$$\xi =0.85{\left({\displaystyle \frac{T_{\mathrm{c}}}{L}}\right)}^{-0.5}+1.49$$

(19)

$$\sigma =0.13{\left({\displaystyle \frac{S_{\mathrm{d}}}{D}}\right)}^{0.51}+1.39$$

(20)

The fitting effect of Equations (15)–(20) is shown in Figure 18. As can be seen from the figure, according to the fitting calculation, the goodness of fit of the curve is 84.3%, and the fitting effect of the equation is good.

5.3. Bearing Capacity Design

In clay-covered silty soil layers, the VHMT bearing capacity design method for shared suction anchors considering soil scour effects is shown in Figure 19.

(1)

Based on the field survey data and soil report, determine the clay layer thickness ratio T_c/D, the suction anchor length-to-diameter ratio L/D, the normalized clay strength S_um/(kD), the normalized silt cohesion, the scour depth ratio S_d/D, the effective unit weight of clay ${{\gamma }_{\mathrm{c}}}^{\prime }$, the effective unit weight of silt ${{\gamma }_{\mathrm{s}}}^{\prime }$, and the silt friction angle ${\phi }_{\mathrm{s}}$.

(2)

Using Equations (2)–(10), calculate the single-bearing loads V₀, H₀, M₀, and T₀ of the suction anchor.

(3)

Represent H₀ and M₀ from step 2 as H_u and M_u, respectively. Using Equations (11)–(14), obtain the horizontal bearing capacity reduction coefficient H₀/H_u and the bending moment bearing capacity reduction coefficient M₀/M_u of the shared suction anchor under different V/V₀ and T/T₀ loading conditions, and calculate H₀ and M₀ under different V/V₀ and T/T₀ loading conditions.

(4)

Using Equations (15)–(20), obtain the normalized VHMT bearing capacity curve of the shared suction anchor.

6. Conclusions

This paper presents finite element modeling of shared suction anchors in clay-covered silty soil layers, investigating their VHMT bearing capacity under soil scour conditions. The VHMT failure envelope equation for shared suction anchors considering soil scour is derived, and a design procedure for their VHMT bearing capacity is proposed, providing practical engineering design references. Key conclusions are as follows:

(1)

It shows that the failure mechanism of the shared anchor in the foundation without scour is different from that in the condition with scour, and the corresponding bearing capacity decreases significantly.

(2)

The tensional force has a significant effect on H, and the bearing capacity of T has a significant contribution to the stability of the shared suction anchor.

(3)

The failure mechanisms of the shared suction anchor in clay overlying silty sand and corresponding bearing envelopes are proposed to assess the bearing capacity of the shared suction anchor, which provides guidance for its application.